Is it possible to solve the following integral:
$$\lim\limits_{c \to +\infty} \displaystyle\int_{1}^{1/c} \dfrac{\sin u}{u} \, du$$
using "elementary" methods? By "elementary", I mean those methods that do not involve Complex analysis, Lebesgue Integration, etc (basically, anything beyond an elementary first course in Real Analysis, say, from the first six chapters of Baby Rudin).
I've seen many solutions to this integral (seemingly with different bounds, including the Dirichlet integral), but all of them seem to use methods that would generally not be accessible to someone with just a basic real analysis course.